Question

How do you solve `3 log x = 6 - 2x`?

Answer 1

Not sure if it can be solved
If you are really curious about the number, the answer is:

`x=2.42337`

Explanation:

Other than using Newton's method, I am not sure if it is possible to solve this. One thing you can do is prove that it has at exactly one solution.

`3logx=6-2x`
`3logx+2x-6=0`

Set:

`f(x)=3logx+2x-6`

Defined for `x>1`

`f'(x)=3/(xln10)+2`

`f'(x)=(3+2xln10)/(xln10)`

For every `x>1` both the numerator and denominator are positive, so the function is increasing. This means it can only have a maximum of one solution (1)

Now to find all the values of `f(x)` `x>1` means `x in(0,oo)`:

`lim_(x->0^+)f(x)=lim_x->(0^+)(3logx+2x-6)=-oo`

`lim_(x->oo)f(x)=lim_(x->oo)(3logx+2x-6)=oo`

Therefore, `f(x)` can take any real value, including 0, which means that `f(x)=0<=>3logx+2x-6=0` can be a solution at least once (2)

(1) + (2) = (Maximum of one) + (At least one) = Exactly one